The Protestant Cemetary in Ostiense is worth a visit to see the graves of Keats and Shelley. So is the Keats and Shelley museum, next to the Spanish Steps.
See Carravagio in San Luigi dei Francesi church.
Go to the Capuchin Crypts - an amazing museum of the Capuchin monks (the guys who wear hair shirts and whip themselves). Actually really fascinating.
If you're looking for food, even just pizza, only go where there are Italian people eating!
I particularly like running atop the city walls from Ostiense. Lovely way to get the views!
The first time I went to Rome I didn't even see the Colusseum, theres so much there you can drown! The best museum by far is the Capitoline Museum - includes graffiti from when Jesus was crucified and a huge Augustus statue (albeit in pieces!).
I'm also pretty nerdy about the Archeological Museum - one dig of a street takes you back through millenia of history.
If you want to go.anywhere, go early. Get to museums etc as they open.
I also echo pp comments about Ostia Antica. Its an absolute gem. You can go to the Appian Way (where Caeser marched his troops into Rome) and hire a bike and cycle the 8 miles there. Loads of sites on the way too. Don't do.this from May-September as its too damn hot!
I'm blessed in that my brother lives in Rome so I go often. He and his GF like Testaccio and Trastevere for food and jazz.